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Register a new userDilation of a class of quantum dynamical semigroups with unbounded generator on UHF algebras
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Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
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We consider the GNS Hilbert space $mathcal{H}$ of a uniformly hyper-finite $C^*$- algebra and study a class of unbounded Lindbladian arises from commutators. Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution. The vacuum expectation of homomorphic co-cycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian. We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.
Let $x_0$ be a possibly-unbounded self-adjoint random variable, $tildesigma_alpha$ and $sigma_beta$ are semicircular variables with variances $alphageq 0$ and $beta>0$ respectively (when $alpha = 0$, $tildesigma_alpha = 0$). Suppose $x_0$, $sigma_alpha$, and $tildesigma_beta$ are all freely independent. We compute the Brown measure of $x_0+tildesigma_alpha+isigma_beta$, extending the previous computations which only work for bounded self-adjoint random variable $x_0$. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+sigma_{alpha+beta}$. We also compute the example where $x_0$ is Cauchy-distributed.
Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used also in the unbounded case, but rather little is known about the general form of unbounded generators. In this paper we first give a precise description of the standard form in the unbounded case, emphasizing intuition, and collecting and even proving the basic results around it. We also give a cautionary example showing that the standard form must not be read too naively. Further examples are given of semigroups, which appear to be probability preserving to first order, but are not for finite times. Based on these, we construct examples of generators which are not of standard form.
We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.
We investigate the possibility to suppress interactions between a finite dimensional system and an infinite dimensional environment through a fast sequence of unitary kicks on the finite dimensional system. This method, called dynamical decoupling, is known to work for bounded interactions, but physical environments such as bosonic heat baths are usually modelled with unbounded interactions, whence here we initiate a systematic study of dynamical decoupling for unbounded operators. We develop a sufficient decoupling criterion for arbitrary Hamiltonians and a necessary decoupling criterion for semibounded Hamiltonians. We give examples for unbounded Hamiltonians where decoupling works and the limiting evolution as well as the convergence speed can be explicitly computed. We show that decoupling does not always work for unbounded interactions and provide both physically and mathematically motivated examples.
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